Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 032, 32 pages      arXiv:2307.02346      https://doi.org/10.3842/SIGMA.2025.032
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

Bilateral Bailey Lattices and Andrews-Gordon Type Identities

Jehanne Dousse a, Frédéric Jouhet b and Isaac Konan b
a) Université de Genève, 7-9, rue Conseil Général, 1205 Genève, Switzerland
b) Univ Lyon, Université Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, 69622 Villeurbanne, France

Received October 15, 2024, in final form April 11, 2025; Published online April 29, 2025

Abstract
We show that the Bailey lattice can be extended to a bilateral version in just a few lines from the bilateral Bailey lemma, using a very simple lemma transforming bilateral Bailey pairs relative to $a$ into bilateral Bailey pairs relative to $a/q$. Using this and similar lemmas, we give bilateral versions and simple proofs of other (new and known) Bailey lattices, including a Bailey lattice of Warnaar and the inverses of Bailey lattices of Lovejoy. As consequences of our bilateral point of view, we derive new $m$-versions of the Andrews-Gordon identities, Bressoud's identities, a new companion to Bressoud's identities, and the Bressoud-Göllnitz-Gordon identities. Finally, we give a new elementary proof of another very general identity of Bressoud using one of our Bailey lattices.

Key words: Bailey lemma; Bailey lattice; Andrews-Gordon identities; Bressoud identities; $q$-series; bilateral series.

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